*(English)*Zbl 1244.92048

Summary: We analyze the stability of equilibria for a delayed SIR epidemic model, in which population growth is subject to logistic growth in absence of disease, with a nonlinear incidence rate satisfying suitable monotonicity conditions. The model admits a unique endemic equilibrium if and only if the basic reproduction number ${R}_{0}$ exceeds one, while the trivial equilibrium and the disease-free equilibrium always exist.

First we show that the disease-free equilibrium is globally asymptotically stable if and only if ${R}_{0}\u2a7d1$. Second we show that the model is permanent if and only if ${R}_{0}>1$. Moreover, using a threshold parameter ${\overline{R}}_{0}$ characterized by the nonlinear incidence function, we establish that the endemic equilibrium is locally asymptotically stable for $1<{R}_{0}\u2a7d{\overline{R}}_{0}$ and it loses stability as the length of the delay increases past a critical value for $1<{\overline{R}}_{0}<{R}_{0}$. Our result is an extension of the stability results of *J.-J. Wang, J.-Z. Zhang* and *Z. Jin* [Analysis of an SIR model with bilinear incidence rate. Nonlinear Anal., Real World Appl. 11, No. 4, 2390–2402 (2010; Zbl 1203.34136)].

##### MSC:

92D30 | Epidemiology |

34K20 | Stability theory of functional-differential equations |

34K60 | Qualitative investigation and simulation of models |